Fourier Series¶
- Fourier Series are sums of harmonically related sinusoids
f(t)=c0+∑k=1∞(ckcos(kω0t)+dksin(kω0t))
ω0 is the fundamental frequency
When can we represent a signal as sums of harmonic components:
Only periodic signals, all harmonics of ω0 are periodic in T=2π/ω0
The c0 term¶
- The constant c0 is known as the "DC" term
- Constant offset of a signal
- Average vvalue of the signal over one period
- Can be computed by computing the definite integral over a T f is periodic over and dividing by T
Computing the other ck terms¶
f(t)=c1cos(ω0t)+d1sin(ω0t)+c3cos(3ω0t)+d5sin(5ω0t)
ck=2T∫t0+Tt0f(t)cos(2πktT)dt
dk=2T∫t0+Tt0f(t)sin(2πktT)dt
- Why do we need both sin and cos components?
- Allows for representation of both even and odd components of the signal
Practice¶
f(t)=0.8sin(6πt)−0.3cos(6πt)+0.75cos(12πt)
ω0=6π
T=2πω0
T=1/3
What are the Fourier series coefficients for f(t)?
- c0=0
- c1=−0.3
- c2=0.75
- d1=0.8